Complexity comparison of evaluated waveforms

In this section, complexity of the W-OFDM and FC-F-OFDM is studied. In addition to overall performance, computational complexity is an important metric of wave-form processing especially in UE side, where the device is size and power restricted.

The complexity is measured by the number of real multiplications needed per CP-OFDM symbol in transmitter processing. Both waveform processing methods to be evaluated (W-OFDM and FC-F-OFDM) are implemented on top of CP-OFDM, which is based on FFT/IFFT pair, and their complexity is examined first. This is used as a baseline result for waveform specific complexity evaluations.

6.4.1 FC-F-OFDM complexity evaluations

When studying FC-F-OFDM computational complexity, the FFT/IFFT operation complexity is examined first, as it is the core module in both types of filter banks.

For given transform length, FFT and IFFT have the same complexity, so only FFT complexity is considered here. For FFT complexity, the split-radix algorithm is commonly considered to be the most efficient one [55], if the transform length is a power of two. Applying split-radix algorithm, the number of real multiplications in FFT processing is

µ_N =N(log₂(N)−3) + 4, (6.2) whereN is a power of two transform length [55]. The number of real multiplication for the low rate transform used in FC-processing is expressed as

C_OFDM =

M

X

m=1

N_SYMµ_LOFDM,m, (6.3)

where M is the number of subbands and µ_LOFDM,m is the number of real multi-plications required for IFFT of size L_OFDM,m and N_SYM is the number of OFDM symbols. Here, the fullband allocation is assumed and we evaluate the complexity per one symbol over one subband (50 PRBs), meaning that N_SYM = 1 and M = 1.

Consequently, the number of real multiplication per symbol can be expressed as

M_TOT,OFDM =C_OFDM/N_ACT,m, (6.4) where NACT is the number of active subcarriers.

In fullband FC-F-OFDM, FFT size of LOFDM,m = 1024 is used. Hence, the number of real multiplications for the OFDM transmitter processing for one OFDM symbol becomesCOFDM = 1024×(log2(1024)−3) + 4 = 7172applying the Equation (6.2).

The number of blocks needed for the FC Tx processing is determined by TFC-BLOCKS,m =

(L_OFDM,m+L_CP,m)N_SYM+L_O,m−L L_S,m

+ 1, (6.5) where L_OFDM,m and L_CP,m are the OFDM IFFT and CP lengths, respectively, on subband m. L_m is the forward transform size for subband m, L_O,m = λL_m is the number of overlapping samples in FC processing on subbandmwith the overlapping factor of λ, L_S,m = L_m − L_O,m is the corresponding number of non-overlapping samples.

The number of real multiplications required for the FC processing becomes C_FC =T_FC-BLOCKS µ_N +

where µ_Lm and µ_N are the number of multiplications needed for the forward and inverse transforms of length L_m and N, respectively, and k_TB is the number of transition-band weights. For real-valued weights ξ= 2, which is used here.

Finally, the number of multiplications per symbol can be expressed as M_TOT,FC = COFDM+CFC

N_SYMPM

m=1NACT,m

, (6.7)

where N_ACT,m is the number of active subcarriers on subbandm.

Complexity evaluations are performed over one subband (m = 1), which is in this case 50 PRB. LTE parameters listed in Table 5.1 are used here i.e. N_ACT,m = 600, L_OFDM,m = 1024 and L_CP,m = 72. Even though the complexity is evaluated per one symbol, the number of symbols is set to N_SYM = 1000 for FC-F-OFDM evaluations in order to model continuous transmission. This has a significant effect to the complexity results due to the overlapping processing of FC-F-OFDM. In Section 5.2.3, transition bandwidth and overlapping factor are set to k_TB = 3 and λ= 1/2. Transform lengths (L_m and N) are 1024, and thus,µ_Lm =µ_N = 7172 real

multiplications are required with split-radix algorithm, as calculated earlier in this section.

Using aforementioned parameters, the total number of real multiplications in FC-F-OFDM transmitter processing becomes M_TOT,FC ≈ 63.2 per symbol. For conventional OFDM, the corresponding result is M_TOT,OFDM = 7172/600 ≈ 11.9, meaning that for each symbol the FC-F-OFDM processing has around five times higher complexity.

6.4.2 W-OFDM complexity evaluations

In W-OFDM transmitter processing, the number of multiplications is simple to evaluate. Windowing is done in time domain, which means that window size N_ws (in samples) equals the number of additional complex multiplications needed for one symbol. Windowing is performed in both edges of the symbol meaning that N_ws/2 samples are windowed in both edges of the symbol. Each windowed sample is multiplied with the corresponding window value as illustrated in Figure 6.9. For simpler illustration, the window length of N_ws = 10 is chosen and only the right side of the windowed symbol is presented as the windowing is symmetrical in both symbol edges.

Nws/2=5

. . . . . .

Windowing

. . .

W-OFDM symbol CP-OFDM symbol

Window

Figure 6.9 W-OFDM sample wise complex multiplications in windowing processing.

In order to evaluate the number real multiplications, the number of complex multiplication is doubled, as the real and imaginary parts of the symbol is windowed separately. Hence, the total number of multiplications in W-OFDM transmitter processing becomes

M_TOT,W = C_OFDM + 2C_W N_SYMPM

m=1N_ACT,m, (6.8)

where CW is the number of complex multiplications originated from the windowing operation.

In W-OFDM case, only one symbol i.e. NSYM = 1 as the complexity is not proportional to number of symbols. Similar to the FC-F-OFDM, the W-OFDM complexity is evaluated over 1 PRB (M = 1 and NACT,m = 600) and theCOFDM = 7172. The window size used in this thesis is chosen to Nws = 36 (see Section 5.2.2), and thus, the number of additional complex multiplications isCW = 36. Finally, the total number of real multiplications becomes MTOT,W ≈ 12.1. The number of real multiplication of W-OFDM transmitter processing is compared against conventional OFDM and FC-F-OFDM (studied in Section 6.4.1) in Table 6.11.

Table 6.11 Complexity comparison of enhanced OFDM techniques against plain CP-OFDM without channel filtering.

Plain CP-OFDM W-OFDM FC-F-OFDM

11.9 12.1 63.2

Significant difference between enhanced OFDM waveforms can be seen in terms of complexity. The W-OFDM transmitter processing does not increase the complex-ity significantly, whereas the FC-F-OFDM transmitter processing needs five times more real multiplications than plain CP-OFDM without channel filtering. The FC-F-OFDM processing complexity is still significantly smaller than with direct time domain filtering implementation achieving similar level of spectral containment [40].

Furthermore, FC-F-OFDM processing provides flexibility in allocation granularity that is not easily managed with time domain filter applications. Due to the com-plexity and depending on the forthcoming 3GPP 5G NR inband and out-of-band emission requirements it is expected that WOLA can be used in most cases, but FC-F-OFDM should be evaluated as the 2nd generation implementation solution for 5G NR basestations and possibly in user equipment at some point [40]. Fur-thermore, FC-F-OFDM processing provides flexibility in allocation granularity that is not easily managed with time domain filter applications. Due to the complexity and depending on the forthcoming 3GPP 5G NR inband and out-of-band emission requirements it is expected that WOLA can be used in most cases, but FC-F-OFDM should be evaluated as the 2nd generation implementation solution for 5G NR bases-tations and possibly in user equipment at some point.